Imagine an underdefined equation $Ax = b$ where the number of equations is less than the number of variables (unknowns), where $x \in \mathbb{R}^d$.
We know that this system has either no or infinitely many solutions. Let's suppose that it's the latter (infinite solutions). I am having difficulty imagining the spread of the solutions in $\mathbb{R}^d$ and wondering if anyone has good ideas on it. Are there clusters of solution points in this space or they are all spread apart in different parts of the space? My guess is that it's the latter, but not sure what is the best way to show it.
First consider the case when $b = 0$.
If $x = u$ and $x = v$ are two solutions to the equation $Ax = 0$, and $a, b$ are two real numbers, then $$A(au+bv) = a(Au) + b(Av) = a\, 0 + b\, 0 = 0$$ Thus $au + bv$ is also a solution to the equation. Since $a, b$ were arbitrary, that means the set of solutions $V = \{x\in \Bbb R^d\mid Ax = 0\}$ is closed under linear combinations. Thus it must be a vector subspace of $\Bbb R^d$. If we can find $n$ independent vectors $\{v_1, \dots, v_n\} \subset V$, and if any greater set of vectors is not independent, then for every $v \in V$, $$v = a_1v_1 + \dots + a_nv_n$$ for uniquely determined values of $(a_1, \dots, a_n) \in \Bbb R^n$. That is, $V$ looks like $\Bbb R^n$, an $n$-dimensional vector space. Since no set of more than $d$ vectors in $\Bbb R^d$ can be linearly independent, we know that $n \le d$. So $V$ is some subspace of $\Bbb R^d$ of dimension $n$.
Now what happens when $b \ne 0$? Let $x = v_0$ be some fixed solution to $Av_0 = b$. If $v$ is another solution, then $A(v - v_0) = Av - Av_0 = b - b = 0$. So $x = v - v_0$ is a solution to $Ax = b$. And vice versa. If $Au = 0$, then $x = u + v_0$ is a solution to $Ax = b$. So the solution set to $Ax = b$ is just the solution set of $Ax = 0$ translated by $v_0$. Instead of a vector subspace of $\Bbb R^d$, we get a translation of such a space. This is called an "affine subspace" of $\Bbb R^d$.
When $d = 3$, the vector subspaces of $\Bbb R^3$ consist of the entire space $\Bbb R^3$, the origin $\{0\}$, and any lines or planes passing through the origin. The affine subspaces consist of the entire space $\Bbb R^3$, any singleton point $\{p\}$, and any lines or planes, regardless of whether they pass through the origin.